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Solve -x² - x + 2 = 0

The solutions of -x² - x + 2 = 0 are x = -2 or x = 1.

x² +x += 0
−x2−x+2=0⇒x=−2, 1-x^2 - x + 2 = 0 \Rightarrow x = -2,\ 1
Discriminant D = 9 · x ≈ -2, 1

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = -x² - x + 2-4-3-2-1123-14-12-10-8-6-4-224x = -2x = 1vertexy = -x² - x + 2
Graph of y = -x² - x + 2. The curve crosses the x-axis at x = -2 and x = 1; the vertex is at (-0.5, 2.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = -1, c = 2.

    −x2−x+2=0-x^2 - x + 2 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−1)2−4⋅(−1)⋅2=1+8=9D = (-1)^2 - 4\cdot(-1)\cdot2 = 1 + 8 = 9
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x+2)(x−1)=0-\left(x + 2\right)\left(x - 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+2=0⇒x=−2x−1=0⇒x=1x + 2 = 0 \Rightarrow x = -2\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±9−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{9}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−2orx=1x = -2\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -2 back in: a·x² + b·x + c = -1·(4) + (-1)·(-2) + 2 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - x + 2 = 0\ \Rightarrow\ x = -2,\ 1 \]
\[ -x^2 - x + 2 = 0 \]
\[ D = (-1)^2 - 4\cdot(-1)\cdot2 = 1 + 8 = 9 \]
\[ -\left(x + 2\right)\left(x - 1\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{9}}{-2} \]
\[ x = -2\quad\text{or}\quad x = 1 \]

The parabola y = -x² - x + 2

xy
-3.5-6.75
-2.5-1.75
-1.51.25
-0.52.25
0.51.25
1.5-1.75
2.5-6.75

Questions

What are the solutions of -x² - x + 2 = 0?

The solutions of -x² - x + 2 = 0 are x = -2 or x = 1.

Does -x² - x + 2 = 0 factor?

Yes: -1(x+2)(x-1) = 0.

What is the vertex of y = -x² - x + 2?

(-1/2, 9/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = -2.

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